Respuesta :
We will investigate composite bodies of various types.
We have a rectangular prism with the following dimensions:
[tex]\begin{gathered} \text{Length ( L ) = 1}\frac{1}{2}\text{ = 1.5 ft} \\ \\ \text{Width ( w ) = 1 foot} \\ \\ \text{Height ( h ) = 2}\frac{1}{2}\text{ = 2.5 ft} \end{gathered}[/tex]We are to fill an empty rectangular prism with ( n ) number of small cubes with dimension as follows:
[tex]\text{Side length ( a ) = }\frac{1}{2}ft\text{ = 0.5 ft}[/tex]We will determine the volume occupied by ( n ) number of cubes. The general formula for the volume of a cube is as follows:
[tex]\text{Volume ( cube ) = a}^3[/tex]Then for ( n ) number of cubes the volume occupied is:
[tex]\begin{gathered} \text{Volume ( n cubes ) = n}\cdot a^3 \\ \text{Volume ( n cubes ) = n}\cdot0.5^3 \\ \\ \text{Volume ( n cubes ) = n}\cdot\frac{1}{8}ft^3 \end{gathered}[/tex]The volume of an empty rectangular prism is defined as follows:
[tex]\text{Volume ( rectangular prism ) = L}\cdot w\cdot h[/tex]Using the given dimensions we can compute the volume of the prism as follows:
[tex]\begin{gathered} \text{Volume ( rectangular prism ) = ( 1.5 ) }\cdot\text{ ( 1 ) }\cdot\text{ ( 2.5 )} \\ Volume(rectangularprism)=3.75ft^3\text{ } \\ \end{gathered}[/tex]The ( n ) number of small cubes must occupy the entire volume of the rectangular prism. So we will go ahead and equate the volumes of each as follows:
[tex]\text{Volume ( rectangular prism ) = Volume ( n cubes )}[/tex]Plug in the respective expressions and solve for the variable ( n ) as follows:
[tex]\begin{gathered} n\cdot\frac{1}{8}\text{ = 3.75} \\ \\ n\text{ = 30 cubes} \end{gathered}[/tex]Therefore, we can pack 30 cubes into the rectangular prism!
